Average Error: 26.0 → 25.9
Time: 15.5s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le 1.728464397415736153089988456136233162691 \cdot 10^{95}:\\ \;\;\;\;\frac{\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.re \le 1.728464397415736153089988456136233162691 \cdot 10^{95}:\\
\;\;\;\;\frac{\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r69022 = x_re;
        double r69023 = y_re;
        double r69024 = r69022 * r69023;
        double r69025 = x_im;
        double r69026 = y_im;
        double r69027 = r69025 * r69026;
        double r69028 = r69024 + r69027;
        double r69029 = r69023 * r69023;
        double r69030 = r69026 * r69026;
        double r69031 = r69029 + r69030;
        double r69032 = r69028 / r69031;
        return r69032;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r69033 = y_re;
        double r69034 = 1.7284643974157362e+95;
        bool r69035 = r69033 <= r69034;
        double r69036 = x_re;
        double r69037 = r69036 * r69033;
        double r69038 = x_im;
        double r69039 = y_im;
        double r69040 = r69038 * r69039;
        double r69041 = r69037 + r69040;
        double r69042 = 1.0;
        double r69043 = r69033 * r69033;
        double r69044 = r69039 * r69039;
        double r69045 = r69043 + r69044;
        double r69046 = sqrt(r69045);
        double r69047 = r69042 / r69046;
        double r69048 = r69041 * r69047;
        double r69049 = r69048 / r69046;
        double r69050 = r69036 / r69046;
        double r69051 = r69035 ? r69049 : r69050;
        return r69051;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if y.re < 1.7284643974157362e+95

    1. Initial program 23.0

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt23.0

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    4. Applied associate-/r*22.9

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
    5. Using strategy rm

    6. Applied div-inv23.0

      \[\leadsto \frac{\color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    if 1.7284643974157362e+95 < y.re

    1. Initial program 39.0

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt39.0

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    4. Applied associate-/r*39.0

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    5. Taylor expanded around inf 38.3

      \[\leadsto \frac{\color{blue}{x.re}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification25.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le 1.728464397415736153089988456136233162691 \cdot 10^{95}:\\ \;\;\;\;\frac{\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))